The double Gromov-Witten invariants of Hirzebruch surfaces are piecewise polynomial

TL;DR

This paper introduces double Gromov-Witten invariants for Hirzebruch surfaces, demonstrating their piecewise polynomial nature, parity properties, and degree, using floor diagrams and Ehrhart theory.

Contribution

It defines a new class of invariants for Hirzebruch surfaces and proves their piecewise polynomiality and parity properties, extending the understanding of Gromov-Witten invariants.

Findings

Invariants are piecewise polynomial functions.

Each polynomial piece is either even or odd.

The degree of each polynomial piece is computed.

Abstract

We define the double Gromov-Witten invariants of Hirzebruch surfaces in analogy with double Hurwitz numbers, and we prove that they satisfy a piecewise polynomiality property analogous to their 1-dimensional counterpart. Furthermore we show that each polynomial piece is either even or odd, and we compute its degree. Our methods combine floor diagrams and Ehrhart theory.